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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.5.63a
Chapter 2, Problem 2.5.63a

Determine whether the following statements are true and give an explanation or counterexample.


a. The graph of a function can never cross one of its horizontal asymptotes.

Verified step by step guidance
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Step 1: Understand the concept of horizontal asymptotes. A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as the input (x) either goes to positive or negative infinity.
Step 2: Consider the behavior of rational functions. For example, the function f(x) = \(\frac{x}{x^2 + 1}\) has a horizontal asymptote at y = 0.
Step 3: Analyze the graph of f(x) = \(\frac{x}{x^2 + 1}\). As x approaches infinity or negative infinity, the function approaches the horizontal asymptote y = 0. However, for finite values of x, the function can cross the horizontal asymptote.
Step 4: Provide a counterexample. For instance, the function f(x) = \(\frac{x}{x^2 + 1}\) crosses the horizontal asymptote y = 0 at x = 0, since f(0) = 0.
Step 5: Conclude that the statement is false. A function can indeed cross its horizontal asymptote, as demonstrated by the counterexample.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Horizontal Asymptotes

A horizontal asymptote is a line that a graph approaches as the input values (x) approach positive or negative infinity. It indicates the behavior of a function at extreme values. For example, the function f(x) = 1/x has a horizontal asymptote at y = 0, meaning as x becomes very large or very small, the function values get closer to 0.
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Behavior of Functions Near Asymptotes

Functions can behave differently near their asymptotes. While a function may approach a horizontal asymptote, it is not restricted from crossing it at finite values of x. For instance, the function f(x) = sin(x)/x approaches the horizontal asymptote y = 0 as x approaches infinity, but it crosses the x-axis multiple times.
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True/False Statements in Mathematics

In mathematics, determining the truth of a statement often requires understanding definitions and properties. A statement can be true in some contexts and false in others. In this case, the assertion that a function cannot cross its horizontal asymptote is false, as demonstrated by functions that do cross their asymptotes at finite points.
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Related Practice
Textbook Question

Complete the following sentences in terms of a limit.


a. A function is continuous from the left at a if _____.

Textbook Question

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2^− h(x)

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Textbook Question

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1^− f(x)

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Textbook Question

Determine the following limits.


a. limx→2+1x(x−2){\(\displaystyle\]\lim\)_{x\(\to\)2^{+}}}\(\frac{1}{\sqrt{x\left(x-2\right)}\)}

Textbook Question

The hyperbolic cosine function, denoted cosh(x)\(\cosh\]\left\)(x\(\right\)), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh(x)=ex+e−x2\(\cosh\]\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}.


a. Determine its end behavior by analyzing limx→∞cosh(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{\(\cosh\)(x)}} and limx→−∞cosh(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{\(\cosh\)(x)}}.

Textbook Question

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.


a. lim x→−2^+ f(x)